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Runge 2nd Order Method http://numericalmethods.eng.usf.edu
Mechanical Engineering Majors Authors: Autar Kaw, Charlie Barker Transforming Numerical Methods Education for STEM Undergraduates 4/16/2017
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Runge-Kutta 2nd Order Method http://numericalmethods.eng.usf.edu
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Runge-Kutta 2nd Order Method
For Runge Kutta 2nd order method is given by where
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Heun’s Method Heun’s method resulting in where
x y xi xi+1 yi+1, predicted yi Here a2=1/2 is chosen resulting in where Figure 1 Runge-Kutta 2nd order method (Heun’s method)
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Midpoint Method Here resulting in where
is chosen, giving resulting in where
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Ralston’s Method Here is chosen, giving resulting in where
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How to write Ordinary Differential Equation
How does one write a first order differential equation in the form of Example is rewritten as In this case
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Example A solid steel shaft at room temperature of 27°C is needed to be contracted so it can be shrunk-fit into a hollow hub. It is placed in a refrigerated chamber that is maintained at −33°C. The rate of change of temperature of the solid shaft q is given by Find the temperature of the steel shaft after 24 hours. Take a step size of h = seconds.
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Solution Step 1: For
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Solution Cont Step 2:
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Solution Cont The solution to this nonlinear equation at t=86400s is
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Comparison with exact results
Figure 2. Heun’s method results for different step sizes
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Effect of step size Table 1. Effect of step size for Heun’s method
86400 43200 21600 10800 5400 −58466 −2.7298×1021 −24.537 −25.785 −26.027 58440 2.7298×1021 −1.5619 − − 223920 1.0460×1011 5.9845 1.2019 (exact)
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Effects of step size on Heun’s Method
Figure 3. Effect of step size in Heun’s method
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Comparison of Euler and Runge-Kutta 2nd Order Methods
Table 2. Comparison of Euler and the Runge-Kutta methods Step size, h Euler Heun Midpoint Ralston 86400 43200 21600 10800 5400 −153.52 −464.32 −29.541 −27.795 −26.958 −58466 −2.7298×1021 −24.537 −25.785 −26.027 −774.64 − −24.069 −25.808 −26.039 −12163 −19.776 −24.268 −25.777 −26.032 (exact)
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Comparison of Euler and Runge-Kutta 2nd Order Methods
Table 2. Comparison of Euler and the Runge-Kutta methods Step size, h Euler Heun Midpoint Ralston 86400 43200 21600 10800 5400 448.34 737.97 14.426 7.0957 3.5755 122160 5.7292 1.1993 1027.6 76.360 7.0508 1.0707 23844 42.571 6.6305 1.2135 (exact)
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Comparison of Euler and Runge-Kutta 2nd Order Methods
Figure 4. Comparison of Euler and Runge Kutta 2nd order methods with exact results.
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Additional Resources For all resources on this topic such as digital audiovisual lectures, primers, textbook chapters, multiple-choice tests, worksheets in MATLAB, MATHEMATICA, MathCad and MAPLE, blogs, related physical problems, please visit
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THE END
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